Nonequilibrium phase transition of a one dimensional system reaches the absorbing state by two different ways

We study the nonequilibrium phase transitions from the absorbing phase to the active phase for the model of disease spreading (Susceptible-Infected-Refractory-Susceptible (SIRS)) on a regular one dimensional lattice. In this model, particles of three species (S, I and R) on a lattice react as follows: $S+I\rightarrow 2I$ with probability $\lambda$, $I\rightarrow R$ after infection time $\tau_I$ and $R\rightarrow I$ after recovery time $\tau_R$. In the case of $\tau_R>\tau_I$, this model has been found to has two critical thresholds separate the active phase from absorbing phases \cite{ali1}. The first critical threshold $\lambda_{c1}$ is corresponding to a low infection probability and second critical threshold $\lambda_{c2}$ is corresponding to a high infection probability. At the first critical threshold $\lambda_{c1}$, our Monte Carlo simulations of this model suggest the phase transition to be of directed percolation class (DP). However, at the second critical threshold $\lambda_{c2}$ we observe that, the system becomes so sensitive to initial values conditions which suggests the phase transition to be discontinuous transition. We confirm this result using order parameter quasistationary probability distribution and finite-size analysis for this model at $\lambda_{c2}$. Additionally, the typical space-time evolution of this model at $\lambda_{c2}$ shows that, the spreading of active particles are compact in a behavior which remind us the spreading behavior in the compact directed percolation.14